ISSN 0101-8205 www.scielo.br/cam
A sharp observability inequality for Kirchhoff
plate systems with potentials
∗XU ZHANG1 and ENRIQUE ZUAZUA Departamento de Matemáticas, Facultad de Ciencias Universidad Autónoma de Madrid, 28049 Madrid, Spain 1Key Laboratory of Systems and Control, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, China
E-mails: xuzhang@amss.ac.cn / enrique.zuazua@uam.es
Abstract. In this paper, we derive a sharp observability inequality for Kirchhoff plate equations with lower order terms. More precisely, for any T>0 and suitable boundary observation domains
(satisfying the geometric conditions that the multiplier method imposes), we prove an estimate
with an explicit observability constant for Kirchhoff plate systems with an arbitrary finite number
of components and in any space dimension with lower order bounded potentials.
Mathematical subject classification: Primary: 93B07; Secondary: 93B05, 35B37.
Key words: Kirchhoff plate system, observability constant, Carleman inequalities, potential, Meshkov’s construction.
1 Introduction
Let n ≥ 1 and N ≥1 be two integers. Letbe a bounded domain inRnwith C4boundaryŴ,Ŵ0be a nonempty open subset ofŴ, and T > 0 be given and sufficiently large. Put Q=△ (0,T)×,6 =△ (0,T)×Ŵand60=△ (0,T)×Ŵ0. For simplicity, we will use the notation yi = ∂∂xy
i,where xi is the i -th coordinate
#690/06. Received: 01/IX/06. Accepted: 01/X/06.
∗The work is supported by the Grant MTM2005-00714 of the Spanish MEC, the DOMINO
of a generic point x =(x1,∙ ∙ ∙ ,xn)inRn. Throughout this paper, we will use C = C(T, , Ŵ0) and C∗ = C∗(, Ŵ0) to denote generic positive constants depending on their arguments which may vary from line to line.
Set
Y =△ ny ∈ H3()
y|Ŵ =1y|Ŵ =0 o
.
We consider the followingRN-valued plate system with a potential a
∈L∞(0,T; Lp(;RN×N))for some p∈ [n/3,∞]:
yt t +12y−1yt t+ay =0 in Q,
y =1y =0 on6,
y(0)=y0, y
t(0)=y1 in,
(1)
where y=(y1,∙ ∙ ∙,yN)⊤, and the initial datum(y0,y1)is supposed to belong
toH =△ ϕ ∈ H3() ϕ|
Ŵ = 1ϕ|Ŵ = 0
N
× H2()TH1 0()
N
, the state space of system (1). It is easy to show that system (1) admits one and only one weak solution y∈C([0,T];H).
In what follows, we shall denote by|∙|,||∙||pand|||∙|||pthe (canonical) norms
onRN, L∞(0,T;Lp(
;RN×N))and L∞(0,T;W1,p(
;RN×N)), respectively.
We shall study the observability constant K(a)of system (1), defined as the smallest (possibly infinite) constant such that the following observability estimate for system (1) holds:
||(y0,y1)||2H
△
= ||y0||2(H3())N + ||y
1
||2(H2())N
≤ K(a) Z
60
∂y ∂ν 2
+ ∂∂νyt
2
+ ∂1∂νy
2
d xdt, ∀(y0,y1)∈H.
(2)
plate only if(, Ŵ0,T) satisfies suitable conditions, i. e. Ŵ0 needs to satisfy certain geometric conditions and T needs to be large enough.
Obviously the observability constant K(a) in (2) not only depends on the potential a, but also on the domains andŴ0 and on the time T . The main purpose of this paper is to analyze only its explicit and sharp dependence on the potential a.
The main tools to derive the explicit observability estimates are the so-called Carleman inequalities. Here we have chosen to work in the spaceHin which
Carleman inequalities can be applied more naturally. But some other choices of the state space are possible. For example, one may consider similar problems in state spaces of the form(H01())N ×(L2())N or(H2()∩H01())N × (H1
0())
N where the Kirchhoff plate system is also well posed. But the
cor-responding analysis on the observability constants, in turn, is technically more involved.
One of the key points to derive inequality (2) for system (1) is the possibility of decomposing the Kirchhoff plate operator∂t2+12−∂t21as follows:
∂t2+12−∂t21=(∂t t −1)(I −1)+1, (3)
where I is the identity operator. Actually, we set
z= y−1y, (4)
where y is the solution of (1). By the first equation of (1) and noting (3), it follows that
−ay= yt t+12y−1yt t =(∂t t −1)(y−1y)+1y =zt t −1z+y−z.
Therefore the Kirchhoff plate system (1) can be written equivalently as the fol-lowing coupled elliptic-wave system
1y+z−y =0 in Q,
zt t −1z+y−z+ay =0 in Q,
y =z =0 on6,
z(0)=y0−1y0, zt(0)=y1−1y1 in.
Consequently, in order to derive the desired observability inequality for sys-tem (1), it is natural to proceed in cascade by applying the global Carleman estimates to the second order operators in the two equations in system (5). We refer to [2, 3] for related works on Carleman inequalities for other cascade sys-tems of partial differential equations.
Similar (boundary and/or internal) observability problems (in suitable spaces) have been considered for the heat and wave equations in [1], and for the Euler-Bernoulli plate equations in [5]. According to [1] and [5], the sharp observability constants for the heat, wave and Euler-Bernoulli plate equations with bounded potentials a (i.e., p= ∞) contain respectively the product of the following two terms (Recall that C∗ =C∗(, Ŵ
0)and C=C(T, , Ŵ0))
H1(T, a)=exp(C∗T||a||∞), H2(T, a)=exp C∗||a||2/3∞
,
W1(T,a)=exp C||a||1/2∞
, W2(T, a)=exp C||a||2/3∞
,
and
P1(T, a)=exp(C∗T|||a|||∞1/2), P2(T,a)=exp C∗||a||1/3∞
.
As explained in [1, 5], the role that each of these constants plays in the observ-ability inequality is of different nature: H1(T, a), W1(T, a)and P1(T,a)are the constants which arise when applying Gronwall’s inequality to establish the energy estimates for solutions of evolution equations; while H2(T,a), W2(T,a) and P2(T, a)appear when using global Carleman estimates to derive the observ-ability inequality by absorbing the undesired lower order terms.
It is shown in [1, Theorems 1.1 and 1.2] and [5, Theorem 3] that the above observability constants are optimal for the heat, wave and Euler-Bernoulli plate systems (N ≥2) with bounded potentials, in even dimensions n≥2. The proof of this optimality result uses the following two key ingredients:
1) For the heat and Euler-Bernoulli plate equations, because of the infinite speed of propagation, one can choose T as small as one likes and hence-forth H1(T, a) and P1(T,a) can be bounded above by H2(T, a) and
P2(T,a), respectively for T = O
||a||−∞1/3
and O||a||−∞1/6
enough (because of the finite velocity of propagation), for any finite T , W1(T, a)can bounded by W2(T, a)because the power 1/2 for||a||∞in
W1(T, a), given by the modified energy estimate, is smaller than 2/3, the power for||a||∞in W2(T, a), arising from the Carleman estimate. In this way, for any finite T large enough, one gets an upper bound on the ob-servability constant (for the wave equation) of the order of exp C||a||2/3∞
.
2) Based on the Meshkov’s construction [8] which allows finding potentials and non-trivial solutions for elliptic systems decaying at infinity in a super-exponential way, one can construct a family of solutions (for the heat, wave and Euler-Bernoulli plate equations) with suitable localization properties showing that most of the energy is concentrated away from the observation domain. According to this, the observed energies grow exponentially as exp − ||a||2/3∞
for the wave and heat systems and as exp − ||a||1/3∞
for the Euler-Bernoulli plate ones.
Things are more complicated for the Kirchhoff plate systems under consid-eration. Indeed, on one hand, due to the finite speed of propagation, one has to choose the observability time T to be large enough. On the other hand, a modified energy estimate for the Kirchhoff plate systems (see (10) in Lemma 1 in Section 2) yields a power 1/2 for||a||∞which can not be absorbed by the one, 1/3, arising from the Carleman estimate. To overcome this difficulty, the key observation in this paper is that, although T has to be taken to be large, one can manage to use the indispensable energy estimate only in a very short time interval when deriving the desired observability estimate. However, we do not know how to show the optimality of the observability constant at this moment. Indeed, when proving the optimality, the energy estimate has to be used in the whole time duration[0,T] and this breaks down the concentration effect that Meshkov’s construction guarantees, which is valid only for very small time du-rations for the Kirchhoff plate systems. Therefore, proving the optimality of the observability estimates obtained in this paper is an interesting open problem.
system. In Section 4 we explain more carefully the main difficulty to show the optimality of the observability constant for Kirchhoff plate systems by means of the above mentioned Meshkov’s construction.
2 Preliminaries
In this section, we show some preliminary energy estimates for Kirchhoff plate systems, and weighted pointwise estimates for the wave and elliptic operators. The estimates for the Kirchhoff plate system will then be obtained by noting the equivalence between system (1) and the coupled wave-elliptic system (5).
2.1 Energy estimates for Kirchhoff plate systems
Denote the energy of system (1) by
E(t)= 1 2 h
|1yt(t,∙)|2(L2())N + |yt(t,∙)|2(H1
0())N + |
1y(t,∙)|2 (H1
0())N
i . (6)
Note that this energy is equivalent to the square of the norm inH. For
s0=
n
3 p, (7)
consider also the modified energy function:
E(t)= E(t)+1
2||a||
2 2−s0
p |y(t,∙)|2(L2())N. (8)
It is clear that both energies are equivalent. Indeed,
E(t)≤E(t)≤C
1+ ||a||
2 2−s0
p
E(t). (9)
The following estimate holds for the modified energy:
Lemma 1. Let a ∈ L∞(0,T;Lp(;RN×N))for some p ∈ [n/3,∞]. Then there is a constant C0=C0(,p,n) >0, independent of T , such that
E(t)≤C0eC0||a||
1 2−s0
Proof. For simplicity, we assume N =1. The same proof applies to a system with any finite number of components N . Using (8) and noting system (1), it is easy to see that
dE(t)
dt = −
Z
ay1ytd x + ||a||
2 2−s0
p
Z
yytd x. (11)
Put p1= s 2
0−2 p−1 and p2=
2
1−s0. Noting that
1 p +
1 p1 +
1 p2 +
1
2 =1 and 1
2s0−1 + 1
2(1−s0)−1+ 1 2 =1,
by Hölder’s inequality and Sobolev’s embedding theorem, and recalling (7)–(8) and observing s0p1= s 2s0
0−2 p−1 =
2n
n−6, we get
− Z
ay1ytd x
≤ Z |
a||y|s0|y|1−s0|1y
t|d x
≤ ||a||p
|y(t,∙)|s0
Lp1()
|y(t,∙)|1−s0
Lp2()||1yt(t,∙)||L2()
= ||a||p||y(t,∙)||s0
Ls0 p1()||y(t,∙)|| 1−s0
L(1−s0)p2()||1yt(t,∙)||L2()
= ||a||p||y(t,∙)||s0 L
2n
n−6()||
y(t,∙)||1−s0
L2()||1yt(t,∙)||L2()
=C||a||
1 2−s0
p ||y(t,∙)||s0 Ln2n−6()
| {z }
≤E(t)s02
||a||
1−s0
2−s0
p ||y(t,∙)||1L−2()s0 !
| {z }
≤E(t)
1−s0 2
||1yt(t,∙)||L2()
| {z }
≤E(t)1/2
≤C||a||
1 2−s0
p E(t).
(12)
Similarly,
||a||
2 2−s0
p Z
yytd x
≤ || a|| 1 2−s0 p
2 R
||a||
2 2−s0
p |y|2+ |yt|2
d x
≤ C||a||
1 2−s0
p E(t).
(13)
Now, combining (11)–(13), and applying Gronwall’s inequality, we conclude the
2.2 Pointwise weighted estimates for the wave and elliptic operators
In this subsection, we present some pointwise weighted estimates for the wave and elliptic equations that will play a key role when deriving the sharp observ-ability estimates for the Kirchhoff plate system.
First, we show a pointwise weighted estimate for the wave operator “∂t t−1”.
For this, for any (large)λ >0, any x0∈Rnand c∈R, set
ℓ(t,x)=λ
|x −x0|2−c
t− T 2
2
. (14)
By taking(ai j)n×n= I , the identity matrix, andθ =eℓ(withℓgiven by (14))
in [4, Corollary 4.1] (see also [7, Lemma 5.1]), one has the following pointwise weighted estimate for the wave operator.
Lemma 2. For any u=u(t,x)∈C2(R1+n), any k
∈Randv=△ θu, it holds
θ2|ut t−1u|2+2
h
ℓt(v2t + |∇v|
2
)−2(∇ℓ)∙(∇v)vt −9vvt
+(A+9)ℓtv2
i
t
+2
n
X
i=1 n
2vi(∇ℓ)∙(∇v)−ℓi|∇v|2+9vvi−2ℓtvtvi +ℓivt2
−(A+9)ℓiv2
o
i
≥2λ(1−k)vt2+2λ(k+3−4c)|∇v|2+Bv2, ∀(t,x)∈R1+n, (15)
where
9=△λ(2n−2c−1+k),
A=4λ2
h
c2(t−T/2)2− |x−x0|2
i
+λ(4c+1−k),
B=8λ3h(4c+5−k)|x−x0|2−(8c+1−k)c2(t−T/2)2
i
+O(λ2). (16)
Corollary 1. Let p= p(t,x)∈C2(R1+n), and set q =θp. Then
θ2|1p|2+2
n
X
i=1
n
2qi(∇ℓ)∙(∇q)−ℓi|∇q|2+9eqqi −(eA+e9)ℓiq2
o
i
≥6λ|∇q|2+eBq2, ∀(t,x)∈R1+n,
(17)
where
e
9=△ λ(2n−1), eA= −4λ2|x−x0|2+λ, e
B=40λ3|x−x0|2+O(λ2), uniformly w.r.t. t ∈ [0,T].
(18)
Proof. We fix an arbitrary t ∈ [0,T] and view the corresponding function which depends on x as a function of(x,s)with s being a fictitious time parameter. We then set
U(s,x)≡ p(t,x), V(s,x)=4(x)U(s,x) ∀(s,x)∈R1+n, where4 = eL and L = λ|x −x
0|2. Choosing c = 0 in (14), and applying Lemma 2 (with k=0) in the variable(x,s)to the above U and V , we get
42|1U|2+2
n
X
i=1
n
2Vi(∇L)∙(∇V)−Li|∇V|2+9eV Vi−(eA+9)e LiV2
o
i
≥6λ|∇V|2+eB V2,
(19)
with9,e eA and eB given by (18). Now, for any c∈R, multiplying both sides of
(19) by e−2cλ(t−T2) 2
, noting
θ =4e−cλ(t−T2) 2
, ℓ=L−cλ
t−T 2
2
and q =e−cλ(t−T2) 2
V ,
the desired inequality (17) follows.
In the sequel, for simplicity, we assume x0 ∈ Rn\(For the general case where, possibly, x0 ∈, we can modify an argument in [7, Case 2 in the proof of Theorem 5.1] to derive the same result). Hence
0<R0=△ minx∈|x −x0|< R1=△ maxx∈|x−x0|. (20) Also, for anyβ >0, we set
2=2(t)=△ expn−βR1
t −
βR1
T −t o
, 0<t<T. (21)
It is easy to see that2(t)decays rapidly to 0 as t →0 or t → T . The desired pointwise Carleman-type estimate with singular weight2for the wave operator reads as follows:
Theorem 1. Let u ∈ C2([0,T] ×) andv = θu. Then there exist four constants T0 >0,λ0 > 0,β0 >0 and c0 >0, independent of u, such that for
all T ≥T0,β ∈(0, β0)andλ≥λ0it holds θ22|ut t −1u|2+2
n
2hℓt(v2t + |∇v|
2)
−2(∇ℓ)∙(∇v)vt −9vvt
+(A+9)ℓtv2
io
t
+22
n
X
i=1 n
2vi(∇ℓ)∙(∇v)−ℓi|∇v|2+9vvi −2ℓtvtvi +ℓiv2t
−(A+9)ℓiv2
o
i
≥c0λθ22(u2t + |∇u|
2
+λ2u2),
(22)
with A and9given by(16).
Step 1. We multiply both sides of inequality (15) by2. Obviously, we have (recall (16) for A and9)
2hℓt(vt2+ |∇v|
2)
−2(∇ℓ)∙(∇v)vt −9vvt+(A+9)ℓtv2
i
t
= n
2 h
ℓt(vt2+ |∇v|
2
)−2(∇ℓ)∙(∇v)vt−9vvt+(A+9)ℓtv2
io
t
−βT R1t−2(T −t)−2(T −2t)2 h
ℓt(v2t + |∇v|
2) −2(∇ℓ)∙(∇v)vt−9vvt +(A+9)ℓtv2
i .
(23)
Note that
−βT R1t−2(T −t)−2(T −2t)2 h
−2(∇ℓ)∙(∇v)vt −9vvti
≤βT R1t−2(T −t)−2|T −2t|2 h
2|(∇ℓ)∙(∇v)vt| + |9vtv|
i
≤βT R1t−2(T −t)−2|T −2t|2 h
(|∇ℓ| +1)vt2+ |∇ℓ||∇v|2+1 49
2v2i. (24)
Thus by (15), and using (23)–(24), we get
θ22|ut t −1u|2+2
n 2
h
ℓt(vt2+ |∇v|
2
)−2(∇ℓ)∙(∇v)vt−9vvt
+(A+9)ℓtv2
io
t
+22
n
X
i=1 n
2vi(∇ℓ)∙(∇v)−ℓi|∇v|2+9vvi −2ℓtvtvi +ℓiv2t
−(A+9)ℓiv2
o
i
≥ 22λ(1−k)v2t +22λ(k+3−4c)|∇v|2 +2βT R1t−2(T −t)−2(T −2t)ℓt2(v2t + |∇v|
2) −2βT R1t−2(T −t)−2|T −2t|2
h
(|∇ℓ| +1)vt2+ |∇ℓ||∇v|2 i
+2hB+2βT R1t−2(T −t)−2(T −2t)ℓt(A+9)
−βT R1t−2(T −t)−2|
T −2t|
2 9
2iv2,
(25)
Step 2. Recalling thatℓand9are given respectively by (14) and (16), we get
22λ(1−k)v2t +22λ(k+3−4c)|∇v|2
+ 2βT R1t−2(T −t)−2(T −2t)ℓt2(v2t + |∇v|
2) − 2βT R1t−2(T −t)−2|T −2t|2
h
(|∇ℓ| +1)vt2+ |∇ℓ||∇v|2 i
+ 2 h
B+2βT R1t−2(T −t)−2(T −2t)ℓt(A+9)
− βT R1t−2(T −t)−2|
T −2t|
2 9
2iv2 = λ2(F1vt2+F2|∇v|2)+λ32Gv2,
(26)
where
F1=△ 2(1−k)+2cβT R1t−2(T −t)−2(T −2t)2 −2βT R1t−2(T −t)−2|T −2t|(2|x −x0| +λ−1),
(27)
F2=△ 2(k+3−4c)+2cβT R1t−2(T −t)−2(T −2t)2 −4βT R1t−2(T −t)−2|T −2t||x−x0|
(28)
and
G=△ 8h(4c+5−k)|x−x0|2−(8c+1−k)c2(t−T/2)2 i
+O(λ−1)
+8cβT R1t−2(T −t)−2(T −2t)2 h
c2(t
−T/2)2 −|x −x0|2+ O(λ−1)
i
−β(2n−2c−1+k)2T R1t−2(T −t)−2|t−T/2|λ−1.
(29)
Thus, by (25) and (26), we have
θ22|ut t −1u|2+2
n 2
h
ℓt(vt2+ |∇v|
2
)−2(∇ℓ)∙(∇v)vt−9vvt
+(A+9)ℓtv2
io
t
+22
n
X
i=1 n
2vi(∇ℓ)∙(∇v)−ℓi|∇v|2+9vvi −2ℓtvtvi +ℓiv2t
−(A+9)ℓiv2
o
i
≥λ2(F1v2t +F2|∇v|2)+λ32Gv2.
Step 3. Let us show that F1, F2and G are positive whenλis large enough. For this purpose, we choose c∈(0,1)sufficiently small so that
(4+5c)R02 9c >R
2
1, (31)
and T(>2R1)sufficiently large such that 4(4+5c)R20
9c >c
2T2>4R2
1. (32)
Also, we choose
k =1−c. (33)
By (33) and recalling that c ∈ (0,4/5), it is easy to see that the nonsingular part F10 =△ 2(1−k)of F1(resp. F20
△
=2(k+3−4c)of F1) is positive. Using (33) again, the nonsingular part of G reads
G0=△ 8h(4c+5−k)|x−x0|2−(8c+1−k)c2(t−T/2)2 i
+O(λ−1) ≥2
h
4(4+5c)R02−9c 3
T2 i
+O(λ−1),
which, via the first inequality in (32), is positive provided thatλis sufficiently large.
When t is near 0 and T , i.e., t ∈ I0 =△ (0, δ0)∪(T −δ0,T)for some suffi-ciently smallδ0 ∈ (0,T/2), the dominant terms in Fi (i =1,2) and G are the
singular ones. For t ∈ I0, the singular part of F1reads
F11 =△ 2cβT R1t−2(T −t)−2(T −2t)2
−2βT R1t−2(T −t)−2|T −2t|(2|x−x0| +λ−1)
≥ 2βT R1t−2(T −t)−2|T −2t|[c(T −2δ0)−2R1−λ−1)] = 2βT R1t−2(T −t)−2|T −2t|(cT −2R1−2cδ0−λ−1), which, via the second inequality in (32), is positive provided that bothδ0 and λ−1are sufficiently small. Similarly, for t ∈ I0, the singular part of F2,
F21 =△ 2cβT R1t−2(T −t)−2(T −2t)2
is positive provided thatδ0 is sufficiently small. Also, for t ∈ I0, the singular part of G reads
G1=△ 8 cβT R1t−2(T −t)−2(T −2t)2 h
c2(t−T/2)2− |x−x0|2 +O(λ−1)
i
−β(2n−2c−1+k)2T R1t−2(T −t)−2|t−T/2|λ−1. It is easy to see that, for t∈ I0, it holds
G1
= βT R1t−2(T −t)−2|T −2t|
8c|T −2t|c2(t
−T/2)2
− |x −x0|2+O(λ−1)
−(2n−2c−1+k)2(2λ)−1 = βT R1t−2(T −t)−2|T −2t|
8c|T −2t|c2(t−T/2)2− |x −x0|2
+O(λ−1) ≥ βT R1t−2(T −t)−2|T −2t|
8c|T −2δ0|
c2(δ0−T/2)2−R21
+O(λ−1) ≥ βT R1t−2(T −t)−2|T −2t|
8c|T −2δ0|
c2T2/4 −R2
1+c2δ0(δ0−T)
+O(λ−1) , which, via the second inequality in (32), is positive provided that bothδ0 and λ−1are sufficiently small.
By (27)–(29), we see that F1= F10+F 1
1, F2 =F20+F 1
2 and G=G 0
+G1. Since F0
1, F20and G0are positive, by the above argument, we see that F1, F2and
G are positive for t ∈ I0. For t ∈(0,T)\I0, noting again the positivity of F10,
F0 2 and G
0, one can chooseβ > 0 sufficiently small such that F1 1, F
1 2 and G
1 are very small, hence so that F1, F2 and G are positive. Hence (30) yields the desired (22). This completes the proof of Theorem 1.
Similar to Theorem 1, by multiplying both sides of (17) by2, we have
θ22|1p|2+22 n
X
i=1
n
2qi(∇ℓ)∙(∇q)−ℓi|∇q|2+9eqqi −(eA+e9)ℓiq2
o
i
≥c0λθ22
|∇p|2+λ2p2,
(34)
with eA ande9given by(18).
3 Sharp observability estimate
In this section we establish a sharp observability estimate for system (1). For this purpose, for any fixed x0 ∈ Rn \ (As mentioned before, we do not really need to assume that x0 is out of . Indeed, for the case where, possibly, x0 ∈ , we can modify an argument in [7, Case 2 in the proof of Theorem 5.1] to derive the same observability result in this section), we introduce the following set:
Ŵ0=△
x ∈Ŵ(x−x0)∙ν(x) >0 . (35) One of the main results in this paper is the following observability inequality with explicit dependence of the observability constant on the potential a for system (1):
Theorem 3. Let Ŵ0 be given by (35) and p ∈ [5n/2,∞]. Then there is a
constant C > 0 such that for any T > T0, with T0 as in Theorem 1, and
any a ∈ L∞(0,T;Lp(
;RN×N)), the weak solution y of system(1) satisfies estimate(2)with the observability constant K(a) >0 verifying
K(a) ≤ C exp
C||a||
1 3−5n/2 p
p
. (36)
Q, noting that2(t)decays rapidly to 0 as t → 0+or t → T−, recalling that z|6=0 (and hence∇z = ∂ν∂zνand zi = ∂ν∂zνi on6), one may deduce that
λ
Z
Q
θ22(z2t + |∇z|2)d xdt+λ3
Z
Q
θ22z2d xdt
≤C
Z
Q
θ22(zt t −1z)2d xdt+4λ
Z
6
θ22
∂ν∂z
2(x−x0)∙ν(x)d xdt
≤C
Z
Q
θ22(zt t −1z)2d xdt+eCλ
Z
60
2
∂ν∂z
2d xdt
≤C
Z
Q
θ22[(ay)2+y2+z2]d xdt+eCλ
Z
60
2∂y ∂ν 2 + ∂1∂νy
2 d xdt , (37)
with60=△ (0,T)×Ŵ0andŴ0being given in (35).
Similarly, applying Theorem 2 respectively to y and yt, we deduce that
λ Z
Q
θ22|∇y|2d xdt+λ3 Z
Q
θ22y2d xdt
≤Cn Z Q
θ22(1y)2d xdt+eCλ Z
60
2 ∂∂νy
2
d xdto
≤Cn Z Q
θ22(y2+z2)d xdt+eCλ Z
60
2 ∂∂νy
2
d xdto,
(38)
and
λ Z
Q
θ22|∇yt|2d xdt+λ3
Z
Q
θ22yt2d xdt
≤Cn Z Q
θ22(1yt)2d xdt+eCλ
Z
60
2 ∂∂νyt
2
d xdto
≤C n Z
Q
θ22(yt2+z
2
t)d xdt+e Cλ Z 60 2 ∂∂νyt
2
d xdt o
.
(39)
It is easy to see that the term CRQθ22z2d xdt (resp. CRQθ22y2d xdt and CRQθ22y2
td xdt ) in the right hand side of (37) (resp. (38) and (39)) can be
λ
Z
Q
θ22(z2t + |∇z|2)d xdt+λ3
Z
Q
θ22z2d xdt
+ ε0λ2
Z
Q
θ22|∇yt|2d xdt+ε0λ4
Z
Q
θ22(yt2+ |∇y|2)d xdt
+ ε0λ6
Z
Q
θ22y2d xdt
≤C
n
||θ√2ay||2L2(Q)+ Z
Q
θ22y2d xdt+ε0
Z
Q
θ22(λ3z2+λz2t)d xdt
+ eCλ
Z
60
2∂y ∂ν 2+ ∂∂νyt
2+
∂1∂νy
2d xdt
o
.
(40)
By takingε0>0 sufficiently small (which is independent ofλ), one can absorb the undesired terms Cε0
R
Qθ
22(λ3z2
+λz2t)d xdt in the right hand side of (40) by its left hand side. Then, for this choice ofε0and taking takingλ >0 sufficiently large, one can absorb further the undesired terms CRQθ22y2d xdt in the right hand side of (40). Consequently, we arrive at
λ
Z
Q
θ22(z2t + |∇z|2)d xdt+λ3
Z
Q
θ22z2d xdt
+λ2
Z
Q
θ22|∇yt|2d xdt+λ4
Z
Q
θ22(yt2+ |∇y|2)d xdt
+λ6
Z
Q
θ22y2d xdt
≤C
n
||θ√2ay||2L2(Q)+e
Cλ
Z
60
2∂y ∂ν 2+ ∂∂νyt
2+
∂1∂νy
2 d xdt o . (41)
Recalling z= y−1y, one has
λ Z
Q
θ22(z2t + |∇z|2)d xdt+λ3 Z
Q
θ22z2d xdt
≥λ Z
Q
θ22[(1yt −yt)2+ |∇1y− ∇y|2]d xdt
+λ3 Z
Q
θ22(1y−y)2d xdt
≥ λ 2
Z
Q
θ22[(1yt)2+ |∇1y|2]d xdt+
λ3 2
Z
Q
θ22(1y)2d xdt
−λ Z
Q
θ22(yt2+ |∇y|2)d xdt−λ3 Z
Q
θ22y2d xdt.
Combining (41) and (42), it follows
λ
Z
Q
θ22[(1yt)2+ |∇1y|2]d xdt+λ2
Z
Q
θ22|∇yt|2d xdt +λ3
Z
Q
θ22(1y)2d xdt+λ4
Z
Q
θ22|∇y|2d xdt
+λ6
Z
Q
θ22y2d xdt
≤C
n
||θ√2ay||2L2(Q)+eCλ Z
60
2∂y ∂ν 2+ ∂∂νyt
2+
∂1∂νy
2d xdt
o
. (43)
Now we have to get rid of the term||θay||2L2(Q).
By the proof of [1, Theorem 2.2], for anyε >0, we have
||θ√2ay||2L2(Q) ≤ ελ||θ
√
2y||2L2(0,T;H1 0())
+ε−n/(p−n)||a||2 p/(p−n)
p λ−n/(p−n)||θ
√
2y||2L2(Q).
By takingεsmall enough the first termελ||θ√2y||2L2(0,T;H1 0())
can be absorbed by the left hand side of (43). Then, for this choice ofεand takingλsufficiently large, the term
Cε−n/(p−n)||a||2 p/(p p−n)λ−n/(p−n)||θ√2y||2L2(Q)
can be absorbed byλ6RQθ22y2d xdt . For this, we chooseλsuch that Cε−n/(p−n)||a||2 p/(p p−n)λ−
n/(p−n) ≤ 1
2λ 6, which yieldsλ≥C||a||2 p/(6 pp −5n)=C||a||
1/(3−5n/2 p)
p .
Therefore, recalling the definition of E(t)in (6), it follows from (43) that Z T
0
2E(t)dt
≤Cexp
C||a||
1 3−5n/2 p
p Z 60 ∂y ∂ν 2 + ∂∂νyt
2 + ∂1∂νy
2
d xdt.
(44)
Now, for||a||psufficiently large, put
t0= 1 2||a||
1
3−5n/2 p−2−n1/3 p
Hence
2||a||
1 2−n/3 p
p t0= ||a||
1 3−5n/2 p
p . (46)
Recall that p>5n/2. Hence t0is small when||a||pis large. Therefore, by the
definition of2in (21), it is obvious that Z T
0
2E(t)dt ≥ Z 2t0
t0
2E(t)dt ≥ 1 Ce
−Ct0−1
Z 2t0
t0
E(t)dt. (47)
By (9) and Lemma 1 (with s0= 3 pn), it follows that Z 2t0
t0
E(t)dt ≥ 1
1+ ||a||
2 2−n/3 p
p
E(0) Z 2t0
t0
e−C||a||
1 2−n/3 p
p tdt
≥ t0
1+ ||a||
2 2−n/3 p
p
E(0)e−2C||a||
1 2−n/3 p
p t0.
(48)
Combining (45)–(48), and noting (46), it follows Z T
0
2E(t)dt ≥ E(0)exp
−C||a||
1 3−5n/2 p
p
. (49)
Finally, the desired estimates (2) and (36) follow from (44) and (49).
4 An open problem on the optimality of the observability constant for Kirchhoff plate systems
In [5, Theorem 3], it is shown that when p = ∞ the observability constant P2(T,a) for the Euler-Bernoulli plate systems with at least two equations in even space dimensions n ≥ 2 is optimal in what concerns the dependence on the potential a. The main idea to prove this optimality result is the same as that in [1], which is based on a suitable construction of u and q satisfying the following bi-Laplacian equation:
12u =qu, inRn, (50)
Lemma 3. Let n≥2 be even. Then there exist two nontrivial complex-valued functions:
u ∈C∞(Rn; C), q∈C∞(Rn; C)\L∞(Rn; C)
such that(50)is satisfied, and for some constant C:
|u(x)| + |∇u(x)| + |∇1u(x)| ≤Ce−|x|4/3, ∀x ∈Rn. (51)
One may expect that Lemma 3 can be applied to establish a similar optimality result for the observability constant K(a)for the Kirchhoff plate systems as well. However, this is an open problem. We now explain why the above Meshkov’s construction does not seem to suffice for Kirchhoff plate systems. Based on the construction of u and q in Lemma 3, by suitable scaling and localization arguments, one can find a family of rescaled potentials aR(x)= R4q(Rx)with
an L∞-norm of the order of R4and a family of solutions u
R(x)=u(Rx)of the
corresponding bi-harmonic problem, with a decay of the order of
|uR(x)| ≤Cexp
−R4/3|x|4/3.
Without loss of generality we may assume that the boundaryŴ(and therefore the observation subdomainŴ0) is included in the region|x| ≥ 1. This yields a sequence of solutions of the bi-Laplacian system12uR = aRuR in which the
ratio between total energy and the energy concentrated inŴ0 and the norm of the boundary traces is of the order of exp−R4/3. Taking into account that ||aR||∞∼R4, this ratio turns to be of the order of exp
− ||aR||
1/3
∞
. These so-lutions of the above mentioned bi-Laplacian system can be regarded also as solu-tions of the Kirchhoff plate system for suitable initial data. However, they do not fulfill homogeneous boundary conditions. Therefore, one needs to compensate them by subtracting the solution taking their boundary data and zero initial ones. In turn, one has to show that these solutions are as small as exp− ||aR||
1/3
∞
in the energy spaceH. Due to the infinite speed of propagation, this can be easily
done for the Euler-Bernoulli plate systems during a time interval of the order of T ≤μ||aR||−
1/6
∞ (because it suffices to use the energy estimate, which yields
an exponential growth expT||aR||
1/2
∞
time). However, the same approach fails for Kirchhoff plate systems since, in order that the (boundary) observability estimate for these systems to hold, one needs to take the time T to be large enough. In fact, the key point is that, at this level the energy estimate yields an exponential growth expT||aR||
1/2
∞
for the energy evolution, and it has to be used in the whole time duration[0,T]. This breaks down the concentration effect that Meshkov’s construction guarantees.
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